VIEWED WITH RELATION TO THE METHOD OF INVARIANTS. 
521 
taking the sum of the sinister bands in (a.)* for the value of B when we write 
xY, y in place of u„ u^, u^, u^, u„ 
(0, l)x«+2(0, 2)^7^+(3(0, 3) + (l, 2))xY+{4{0, 4)+2(1, 3)^/ 
+ ( 5 ( 0 , 5)+3(l, 4) + (2, 3))^y+(4(l, 5)+2(2, 4)).ry+(3(2, 5) + (3, 4))xy 
+2(3, b)xf+{4, b)y\ 
The direct process requires the calculation of 
{^ax^+4hx^y-\-^cxy-\-2dxy''+ey%^x^^2'yx^y-\-^^xy-\-4^xy^-\-b■ky^) 
- (5 a + 4 ^x^y + 3 yxy + + 53 /^) ( hx ^ + 2 cx^y + 3 dxy -\-4exy^-\-by), 
each coefficient of which will contain the numerical factor 5 ; so that to reduce the 
Jacobian to its simplest form each coefficient will necessitate the employment of 
additions, subtractions, and a division, instead of additions merely, as when the 
Bezoutic square is employed. For instance, to find the coefficient of x^.y from the 
above expression (a.), we have to calculate 
5 ( 25 ( 0 , 5) + 16(l, 4) + 9(2, 3) + 4(3,,2) + (4, 1)), 
i.e. i(25(0, 5) + (I 6 -l)(l, 4) + (9-4)(2, 3)), 
which IS 5(0, 5)+3(l, 4) + (2, 3), agreeing with what has been found above for the 
value of such coefficient, by a simple process of counting. The same remark will, of 
course, also apply to the computation of the Hessian of F by means of its Bezoutoid. 
(Art. 66 .), This relation between the Bezoutiant and the Jacobian led me to 
inquire whether, as would at first sight appear probable, the Bezoutiant were the 
only lineo-linear quadratic function of {m) variables covariantive to /‘and (p (the word 
lineo-linear being used to denote the form of coefficients, such as those in the 
Bezoutiant, linear in respect of the coefficients in /and the coefficients of ^). If so, 
then there would have existed a method of performing the inverse process of recover- 
ing the Bezoutiant from the Jacobian, almost as simple as that of deriving the 
Jacobian from the Bezoutiant. On investigating the matter, however, I found that 
such is by no means the casef , but that there exists a whole family of independent lineo- 
* Vide art. 62. 
t This might have been concluded immediately from the following observation. Let J, the Jacobian of 
/ and (p, be expressed under the form 
A^x‘^ra-2^ (2m - 2) A, . 1 .y ^ (2m - 2) . ys + . . . + A2,„_2 . 
then we know from the Calculus of Forms, that, D being taken to represent the persymmetrical Determinant 
Ao: 
A^: 
A,; . 
A.; 
A3; . 
A,; 
A3: 
A,: . 
Am-i 
Am; 
J • 
3 Y 2 
